A033289 Odd power perfect numbers: numbers k such that opsigma(k) = 2*k, where opsigma(k) = A033634(k).

6, 264, 45408, 10177920, 9310826880, 27806077440, 25437179036160, 303753589954560, 277875743791011840, 14504815632384, 13269098919960576, 2534919599177957376, 2318960803647990104064

Offset: 1

Author

Yasutoshi Kohmoto

Keywords

• nonn
• more

Comments

If x is OPP and x=2^k*y, gcd(2^k,y)=1, (2^(k+4)+1)/3 is prime, then 4*x*(2^(k+4)+1)/3 is also OPP.

By applying the rule above to a(12) we get that 1772040615644549459607552 is also a term. - Amiram Eldar, Aug 26 2022

Crossrefs

Cf. A000978, A033634.

Programs

Mathematica

f[e_] := If[OddQ[e], e+2, e+1]; fun[p_, e_] := 1 + (p^f[e] - p)/(p^2 - 1); opsigma[1] = 1; opsigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[50000], opsigma[#] == 2*# &] (* Amiram Eldar, Aug 26 2022 *)

Formula

{k: A033634(k) = 2*k}.